Pulsed radar system using optimized transmit and filter waveforms

ABSTRACT

The technology described in this document can be embodied in a radar system that employs pulse compression waveforms. In one aspect, a radar system includes a transmitter device and a receiver device, which are both configured to access a storage device. The storage device is configured to store a first sequence of values and a second sequence of values. The first sequence of values can represent phase values for a transmit waveform. The second sequence of values can represent complex values for a filter waveform. The first and second sequences of values are generated via a joint optimization process. An objective function of the first and second sequences is a weighted sum of metrics indicative of a sidelobe level of a simulated range response and frequency suppression of the transmit and filter waveforms outside and portions inside of a target bandwidth.

PRIORITY CLAIM

This application claims priority to U.S. Provisional Application62/619,733, filed on Jan. 20, 2018, and U.S. Provisional Application62/720,194, filed on Aug. 21, 2018. The contents of each of the aboveapplications are incorporated herein by reference.

TECHNICAL FIELD

This disclosure generally relates to pulsed radar systems that utilizepulse compression using transmit and filter waveforms.

BACKGROUND

Pulsed radar waveforms are designed to increase resolution and reducesidelobes. Using identical transmit and filter waveforms (the matchedcase) is known to maximize SNR for a single target and white noise.However, for other environments and applications, such asclutter-dominated environments, multiple target environments, separationof targets from their multipath reflection, and target identificationfor High Range Resolution (HRR) radars, mismatched transmit and filterwaveforms may be designed to produce optimal results according to atleast four performance figures of merit (FOM).

These four FOMs are 1) sidelobe power minimization, 2) constrained SNRloss, 3) spectral containment of both the transmit and filter waveforms,and 4) Doppler tolerance for moving targets, platforms, or both. Poorperformance in just one of these FOMs can result in a waveformunsuitable for a given application.

While some conventional systems, apparatuses, and methods may addresssubsets of these FOMs, adequate performance for all four is necessaryfor a high performance radar system. Thus, there is a need for systems,apparatuses, and methods for simultaneously optimizing transmit andfilter waveforms with respect to all four FOMs.

SUMMARY

This document describes technology referred to as Mismatched OversampledWaveforms (MOW), in which pulsed radar systems employ pulse compressionusing oversampled mismatched transmit and filter waveforms that aresimultaneously optimized for complex, real-world environments. Acomputing device-implemented method is described in which theoversampled mismatched transmit and filter waveforms are simultaneouslyoptimized so that 1) the cross-correlation of the two waveforms producesa range response with reduced sidelobe power, 2) SNR loss is constrainedto an acceptable level, 3) spectral leakage is contained for both thetransmit and filter waveforms, and 4) performance is robust to a broadrange of Doppler frequencies.

In one aspect, a computing device-implemented method includes receivingone or more input parameters that include a first parameter indicativeof a number of code samples of a first waveform; a second parameterindicative of a number of code samples of a second waveform, in whichthe number of code samples of the second waveform is greater than thenumber of code samples of the first waveform; and a third parameterindicative of an oversampling ratio. The method further includesgenerating, by one or more processing devices executing a jointoptimization process, (i) a first sequence of values representing phasevalues for the first waveform, and (ii) a second sequence of valuesrepresenting a set of complex amplitude values for the second waveform.The first and second sequences of values are generated such that anobjective function of the first sequence and the second sequencesatisfies a set of one or more conditions, subject to one or moreconstraints. The objective function includes a weighted sum of a Dopplertolerant metric indicative of a sidelobe level of a simulated rangeresponse, a metric indicative of a frequency suppression of the firstwaveform inside and/or outside of a target bandwidth, and a metricindicative of a frequency suppression of the second waveform insideand/or outside of the target bandwidth. The method further includesstoring the first sequence of values and the second sequence of valuesin a storage device accessible to a transmitter device and a receiverdevice such that a measured range response to a transmit waveformtransmitted by the transmitter device based on the first sequence ofvalues is computed by the receiver device as a cross-correlation between(i) a sequence of values representing a received signal, and (ii) thesecond sequence of values.

In another aspect, a radar system includes a transmitter deviceconfigured to transmit a transmit signal encoding a transmit waveformhaving a plurality of pulses, and a receiver device configured toreceive a received signal. The transmitter device and the receiverdevice are further configured to access a storage device configured tostore a first sequence of values and a second sequence of values suchthat a measured range response to the transmit signal transmitted by thetransmitter device based on the first sequence of values is computed bythe receiver device as a cross-correlation between (i) a sequence ofvalues representing the received signal, and (ii) the second sequence ofvalues. The first sequence of values and the second sequence of valuesare generated by a computing device configured to execute instructionsto perform operations including receiving one or more input parametersthat include a first parameter indicative of a number of code samples ofa first waveform; a second parameter indicative of a number of codesamples of a second waveform, in which the number of code samples of thesecond waveform is greater than the number of code samples of the firstwaveform; and a third parameter indicative of an oversampling ratio. Theoperations further include generating, by one or more processing devicesexecuting a joint optimization process, (i) the first sequence of valuesrepresenting phase values for the first waveform, and (ii) the secondsequence of values representing a set of complex amplitude values forthe second waveform. The first and second sequences of values aregenerated such that an objective function of the first sequence and thesecond sequence satisfies a set of one or more conditions, subject toone or more constraints. The objective function includes a weighted sumof a Doppler tolerant metric indicative of a sidelobe level of asimulated range response, a metric indicative of a frequency suppressionof the first waveform inside and/or outside of a target bandwidth, and ametric indicative of a frequency suppression of the second waveforminside and/or outside of the target bandwidth.

In another aspect, a system includes a computing device that includes amemory configured to store instructions and a processor to execute theinstructions to perform operations. The operations include receiving oneor more input parameters that include a first parameter indicative of anumber of code samples of a first waveform; a second parameterindicative of a number of code samples of a second waveform, in whichthe number of code samples of the second waveform is greater than thenumber of code samples of the first waveform; and a third parameterindicative of an oversampling ratio. The operations further includegenerating, by one or more processing devices executing a jointoptimization process, (i) a first sequence of values representing phasevalues for the first waveform, and (ii) a second sequence of valuesrepresenting a set of complex amplitude values for the second waveform.The first and second sequences of values are generated such that anobjective function of the first sequence and the second sequencesatisfies a set of one or more conditions, subject to one or moreconstraints. The objective function includes a weighted sum of a Dopplertolerant metric indicative of a sidelobe level of a simulated rangeresponse, a metric indicative of a frequency suppression of the firstwaveform inside and/or outside of a target bandwidth, and a metricindicative of a frequency suppression of the second waveform insideand/or outside of the target bandwidth. The operations further includesstoring the first sequence of values and the second sequence of valuesin a storage device accessible to a transmitter device and a receiverdevice such that a measured range response to a transmit waveformtransmitted by the transmitter device based on the first sequence ofvalues is computed by the receiver device as a cross-correlation between(i) a sequence of values representing a received signal, and (ii) thesecond sequence of values.

In another aspect, one or more computer readable media storeinstructions that are executable by a processing device, and upon suchexecution cause the processing device to perform operations. Theoperations include receiving one or more input parameters that include afirst parameter indicative of a number of code samples of a firstwaveform; a second parameter indicative of a number of code samples of asecond waveform, in which the number of code samples of the secondwaveform is greater than the number of code samples of the firstwaveform; and a third parameter indicative of an oversampling ratio. Theoperations further include generating, by one or more processing devicesexecuting a joint optimization process, (i) a first sequence of valuesrepresenting phase values for the first waveform, and (ii) a secondsequence of values representing a set of complex amplitude values forthe second waveform. The first and second sequences of values aregenerated such that an objective function of the first sequence and thesecond sequence satisfies a set of one or more conditions, subject toone or more constraints. The objective function includes a weighted sumof a Doppler tolerant metric indicative of a sidelobe level of asimulated range response, a metric indicative of a frequency suppressionof the first waveform inside and/or outside of a target bandwidth, and ametric indicative of a frequency suppression of the second waveforminside and/or outside of the target bandwidth. The operations furtherincludes storing the first sequence of values and the second sequence ofvalues in a storage device accessible to a transmitter device and areceiver device such that a measured range response to a transmitwaveform transmitted by the transmitter device based on the firstsequence of values is computed by the receiver device as across-correlation between (i) a sequence of values representing areceived signal, and (ii) the second sequence of values.

Implementations may include one or more of the following features. Theone or more input parameters may be received based on a set of targetcharacteristics for the simulated range response. The set of targetcharacteristics for the simulated range response may include at leastone of a target value for the metric indicative of the sidelobe level ofthe simulated range response, a target value for a metric indicative ofan overall frequency suppression of the simulated range response, and atarget value for a metric indicative of a Doppler tolerance. The metricindicative of the sidelobe level of the simulated range response mayinclude terms that represent at least one of an integrated sidelobelevel, a peak sidelobe level, an average sidelobe level, and a mediansidelobe level. The one or more input parameters may include a samplerate supported by the transmitter device and the receiver device, andmay include a target value for a metric indicative of an overallfrequency suppression of the range response. The one or more constraintsmay include an acceptable SNR loss threshold; a constraint imposed on amainlobe associated with the simulated range response, which may specifythat the mainlobe has a magnitude substantially equal to unity; and/or afirst phase of the first sequence of values constrained to 0. In someimplementations, generating the first sequence of values and the secondsequence of values may include initializing a first set of decisionvariables as a series of phases within a range; initializing a secondset of decision variables as a series of complex values; and generatingthe first and second sequence of values from the first and second set ofdecision variables, respectively, by executing the joint optimizationprocess on the first and second set of decision variables, the jointoptimization process being constrained by the one or more constraints.The series of phases can be a series of random numbers within a 27range. At least one sidelobe of the simulated range response may beexcluded from a computation of the objective function, and in somecases, the at least one excluded sidelobe is directly adjacent to amainlobe associated with the simulated range response. Generating thefirst and second sequence of values may include generating multiplecandidate sets of values, each candidate set comprising a candidatefirst sequence of values and a candidate second sequence of values;computing, for each candidate set, a value of the objective function;determining that a value of the objective function for a particularcandidate set is less than values of corresponding objective functionsof other candidate sets; and responsive to determining that the value ofthe objective function computed for the particular candidate set is lessthan the values of corresponding objective functions of the othercandidate sets, selecting the first and second candidate sequence ofvalues corresponding to the particular candidate set as the first andsecond sequence of values, respectively. The one or more conditions caninclude a condition that a value of the objective function is locallyminimized. The oversampling ratio can be a function of the targetbandwidth and a sample rate. The first set of suppressed frequenciesused to calculate the metric indicative of a frequency suppression ofthe first waveform inside and/or outside of the target bandwidth can bethe same as or different from a second set of suppressed frequenciesused to calculate the metric of a frequency suppression of the secondwaveform inside and/or outside of the target bandwidth. The first set ofsuppressed frequencies and/or the second set of frequencies can includefrequencies that fall within the target bandwidth. The first set ofsuppressed frequencies can be selected to restrict emission to frequencybands that require limited levels of emission. The second set ofsuppressed frequencies can be selected to filter out frequency bands inwhich a strong emitter is emitting. The target bandwidth can bedetermined based on the oversampling ratio and a chip duration. Theamplitudes of the transmit waveform can be ramped up at a start of thetransmit waveform and amplitudes of the transmit waveform can be rampeddown at an end of the transmit waveform. The Doppler tolerant metricindicative of a sidelobe level can be a sum of a time ambiguity functionover a set of Doppler frequencies. For the radar system, the receiverdevice may be further configured to generate an output indicative of alocation of an object as calculated by multi-range processing around thepeak range response.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating an exemplary pulsed radar systemincorporating mismatched oversampled waveforms (MOW) transmit and filtercodes.

FIG. 2 is a flow chart of operations for determining an optimizedtransmit and filter code.

FIG. 3 is a plot comparing sidelobe level reduction for a MOW processcompared to a Cyclic Algorithm New (CAN) algorithm.

FIG. 4 is a plot of MOW range response for varying number of filter codesamples compared with that from the CAN algorithm.

FIG. 5 is a magnified view of range responses from FIG. 4 near amainlobe.

FIG. 6 is a plot of range responses upon excluding sidelobes adjacent tothe mainlobe from the minimization function.

FIG. 7A is a plot of transmit waveform power spectral density foroversampling by L=8, zeroth and first order DACs, 201 transmit samples,and 601 filter samples.

FIG. 7B is a plot of first order hold range response.

FIG. 8 is a flow chart of an example method for storing MOW transmit andfilter codes in a storage device.

DETAILED DESCRIPTION

Radar systems utilize radio waves for object detection, and can beconfigured to determine one or more of the range, angle, and velocity ofobjects. In some cases, this information can be calculated bymulti-range processing around a peak range response of the radar system,processing several range samples for a range estimate. To accomplishobject detection, radar systems can transmit pulses with a phase onlywaveform, as power amplifiers are most efficient and distort less withthis mode. This maximizes radiated energy for a given pulse width for asaturation mode amplifier. To achieve adequate energy on a small radarcross-section (RCS) target, the pulse width may be long, which cansignificantly reduce range resolution. Alternatively, pulse compressionusing waveforms can achieve much better resolution for the same pulsewidth and power. The pulse time dependence is a series of shortsub-pulses or chips, each depending on a complex number. Together, thesenumbers form a code indicative of a waveform.

Conventional pulsed radar systems often use identical transmit andfilter waveforms (known as the matched case), which maximizes SNR inenvironments with a single target and white noise. However, in complexenvironments, it may be more advantageous to use non-identical, ormismatched, transmit and filter waveforms. At least four FOMs relate tothe performance of a radar system that utilizes mismatched transmit andfilter waveforms.

The first two FOMs are sidelobe power minimization and constrained SNRloss. Sidelobe power of a range response may be measured by metricsincluding, but not limited to, the sum of power in all sidelobes knownas Integrated Sidelobe Level (ISL), and the largest sidelobe power knownas Peak Sidelobe Level (PSL). SNR loss in the mismatched case ismeasured relative to that of the matched case.

In the matched case, the range response is an auto-correlation of thetransmit waveform and always results in symmetric outputs. For matchedconstant modulus waveforms of an amplifier operating in a saturatedmode, Linear Frequency Modulation (LFM) chirps have been employed, withimprovements including sections of different chirp rates, hoppedfrequency bursts, more general nonlinear FM, and sinusoidally taperedamplitudes at the first and last thirds of the waveform. Adaptive chirpscan be derived to reduce SNR loss. These chirp modifications oftenbroaden the mainlobe.

Barker codes are unit modulus binary phase codes with lowautocorrelation sidelobes. Longer binary codes and other codes with asmall finite number of phases also have low autocorrelation sidelobes.Polyphase waveforms have continuous phases at each sample. An efficientleast squares algorithm for matched polyphase codes to minimize ISLbased on Fast Fourier Transforms (FFT) is known as Cyclic Algorithm New,or CAN (see, for example, Stoica, He, and Li, 2009; He, Li, and Stoica,2012; Song, 2015), and its ISL is an improvement over that of chirps andfinite phase codes. Additionally, the noise-like waveforms are LowProbability of Intercept (LPI).

Still lower sidelobes are obtained for the mismatched case, where theunit modulus transmit waveform is specified and a complex filterwaveform of more samples and arbitrary amplitude is calculated. In themismatched case, range response is the cross-correlation of the transmitand filter waveforms. Consequently, mismatched codes can have asymmetricweights with resulting asymmetric range and spectral response.

Since it is believed that the specified transmit waveform should have asharply peaked auto-correlation function, binary waveforms such asBarker codes, or other binary codes with good correlation propertiesfound by computer searches, have been adopted as input to derive filtercodes. This calculation can also be extended to multiple transmitwaveforms and chirp transmit waveforms.

In each of these examples, the transmit code is specified based on itsauto-correlation and not based on its suitability with the filterwaveform for producing a range response with desired properties. Oneapproach for reducing ISL is simultaneous calculation of phase onlytransmit codes and mismatched receive filters. These algorithms have anequality constraint for the mainlobe and an inequality constraint thatsets a ceiling to SNR loss. While these algorithms jointly optimize thetransmit and filter waveforms with respect to sidelobe level and SNRloss, they ignore the FOMs of spectral containment and Dopplertolerance. The present technique improves upon these algorithms byfurther optimizing for both spectral containment and Doppler tolerance.

The third FOM is spectral containment. Converting from a discrete codeto continuous time dependence causes spectral spreading. Chip timeduration can be defined as the inverse bandwidth. If the transmit codediscontinuously jumps phases between chips, the resulting time waveformspreads frequency sidelobes outside the allowed bandwidth into otherbands that should not be radiated into, such as those for navigation.Since the receive filter is also not band limited, external sourcestransmitting bands in nearby frequencies will be processed and increasenoise.

Transmit spectral sidelobes can be reduced somewhat by a modulationwhich has no discontinuous phase jumps and generates a continuousintra-pulse time dependence between code samples. This includesDifferential Phase Shift Keying (DPSK) and Minimum Shift Keying (MSK)for binary codes and Continuous Phase Modulation (CPM) for polyphasecodes. CPM has discontinuous first derivatives at chip boundaries andfor some versions, amplitude variability. These modulation algorithmssuppress spectral leakage, but have no explicit frequency penalty.Without this, out-of-band spectral reduction may not satisfytransmission requirements and straddling loss is high. Also, sidelobeminimization is not optimal, as the deterministic oversampled timesamples are not optimized for sidelobe reduction. In the present MOWprocess, these challenges are overcome, with all samples contributing toboth sidelobe reduction and spectral containment, and an explicitfrequency penalty being provided.

Previous techniques oversample each chip by a factor of 2 and solve fora matched filter that minimizes a weighted sum of ISL and a penalty forout-of-band spectral power. The effective low-pass of phase samplesfollowed by CPM produces lowered spectral energy. In anotherapplication, a time waveform is generated from discrete codes by sincinterpolation to reduce spectral leakage, but the resulting waveform isnot unit modulus. Additional samples from CPM smooth code phase jumpsand are not designed to reduce ISL. The present technique oversamples ata much larger factor (nominally 8), the transmit waveform is almostentirely unit modulus, and all samples work to reduce ISL and improvespectral containment, so that a usable emitted waveform is obtained justby a Digital to Analog Converter (DAC) and an analog low-pass filter,and CPM is not needed.

The fourth FOM is Doppler tolerance, which can be important forpolyphase codes, as even a small phase ramp over the uncompressed pulsewidth affects performance, especially for broad band, long pulse codes.Traditionally, a bank of Doppler corrected receive filters has beenproposed but many banks would be required as the Doppler extent foracceptable loss is small. The MOW process addresses this issue byextending an ISL related functional to sum over an input range ofDoppler frequencies to broaden the range of acceptable Dopplerfrequencies.

Unlike the previously mentioned conventional systems and techniques, theMOW process utilizes a joint optimization of oversampled, mismatchedtransmit and filter waveforms to improve radar performance with respectto 1) sidelobe power level, 2) SNR loss, 3) spectral containment, and 4)Doppler tolerance. Further details of the present technique, itsvariations, and its advantages are described herein.

FIG. 1 shows an exemplary monostatic pulsed radar system 100 employingpulse compression waveform codes provided by a storage device 103 (asgraphically represented by lines 117 and 118). The radar system caninclude a transceiver consisting of a transmitting system 115 and areceiving system 116. The transmitting system 115 accepts as input atransmit code from the storage device 103 (as represented by line 117).The transmit code is converted to a time waveform by the Digital toAnalog Converter (DAC) 104, low-pass filtered in block 105, modulated toRF and amplified in 106, sent through the combiner 107 and provided tothe antenna 108. The combiner 107 has a T/R switch allowing it to switchbetween transmit and receive modes. The receiving system 116 receivesreturns via the antenna 108 and the combiner 107, and the returns aredemodulated in the receiver 109. The demodulated returns are then passedthrough an anti-alias filter 110, digitized in the Analog to DigitalConverter (ADC) 111 and correlated (in Correlator 112) with the filtercode (provided by the storage device 103, as represented by line 118),sometimes referred to as a receive waveform or receive filter code. Thepost-processor 113 does Doppler processing, jammer cancellation, trackformation, and other multi-pulse procedures. In some cases, this outputcan guide waveform choice and can be presented on a display 114 of theradar system 100.

The radar system 100 may include an evaluation engine 102 that isconfigured to evaluate completion of a radar mission. The output of theevaluation engine 102 can be based in part from the output of thereceiving system as well as input from an external radar operator. Whilethe evaluation engine is shown as a part of the radar system 100, insome cases, the external radar operator can provide or assist thefunctionality of the evaluation engine 102. Transmit and receivewaveform code pairs are stored in the storage device 103, and based onan output of the evaluation engine 102, a pair of waveforms consistingof a transmit waveform code 117 and a filter code 118 is selected. Forexample, if an extended target is detected, a High Range Resolution(HRR) waveform pair can be selected. The pairs of transmit and filtercodes stored in the storage device 103 can be determined using one ormore techniques described herein.

FIG. 2 shows a flow chart 200 for simultaneously determining a pair ofoptimized transmit and filter waveform codes that may be stored in astorage device 103 of the radar system 100. In general, the techniquerepresented in flowchart 200 optimizes transmit and filter codes byconstrained minimization of a weighted sum of multiple non-linear scalarfunctions, also referred to as the objective function. These functionsinvolve a generalized ISL and spectral content and are explained indetail below. While a constrained minimization of the objective functionis described, in some cases, the objective function may have a form inwhich a constrained maximization is performed.

The flowchart 200 first assigns sidelobe, frequency suppression, andDoppler tolerance requirements 201 and infers input parameters 202,which can include a number of samples, parameters defining processvariations that are shown later, and if the system allows it, a samplerate. Unknown decision variables in the minimization routine aretransmit phases and filter code samples, with decision variables definedas quantities subject to change in order to perform the constrainedoptimization of the objective function.

Once the input parameters are inferred, the simulation loop starts 203.Initial values are assigned 204, the minimization routine starts 205,and a constrained nonlinear minimization routine is performed 206. Forthe purposes of this application, each minimization routine that uses agiven set of range, frequency, Doppler, and scalar weights is referredto as a segment. After an input number of iterations for a particularsegment, the then current solution is used to evaluate the FOMs, ISL,and frequency suppression 207. If the value of the objective functionhas converged or if a maximum number of iterations has been reached in208, the simulation loop is ended 210. If neither the objective functionhas converged nor the maximum number of iterations has been reached, theweights are adjusted 209 and the next segment of iterations is performed205.

In some cases, segment loops are terminated if the sidelobe, frequencysuppression, and Doppler tolerance requirements are satisfied, ifconvergence is obtained, or if a maximum number of iterations isreached. The simulation can be rerun a number of times with differentinitial values to increase the probability that one of the simulationsis close to the global minimum. The best solution is chosen 211, andcodes are output to storage or directly to the transmitter and receiver212. In some cases, the best solution is selected based on a set oftransmit phases and filter code samples that generate the lowest valueof the objective function. While the flowchart 200 represents operationsto perform a minimization routine, it is understood that in some cases,a maximization routine may be implemented.

Implementation of the MOW process is herein described incrementally withrespect to the four FOMs of sidelobe reduction, constrained SNR loss,spectral containment, and Doppler tolerance. First, sidelobeminimization with constrained SNR loss is described for simultaneouslyoptimizing a phase-only transmit and mismatched receive codes. Then, toavoid spectral leakage, an oversampled version is shown, with low passfiltering via frequency band suppression and amplitude ramping of thetransmit waveform. Finally, Doppler tolerance is included.

As described above, a phase-only transmit code and a mismatched complexreceive filter code of more samples are found simultaneously whileminimizing sidelobe power subject to a constrained SNR loss relative tothat of a matched filter. This description is for a single channelantenna and unknown target and clutter spectral and amplitude content,but can be generalized to include Multiple Input Multiple Output (MIMO)radar systems and spectral adaptation.

Correlations are defined for codes of unequal length. Transmit andfilter codes are of length N_(t) and N_(r), respectively, withN_(r)>N_(t). Both N_(t) and N_(r) are odd so that symmetric codes andrange response can be generated. While symmetric codes are preferredbecause they halve the number of unknown decision variables, in somecases, asymmetric codes and range responses can be generated. Thetransmit code is defined asx(n)=b(n)exp[iϕ(n)],n=1, . . . ,N _(t)  (1)where the N_(t) samples have unknown phases φ(n) between −π and πradians and b(n) is a known real amplitude used later for ramping up anddown amplitudes of the transmit waveform. For now, all amplitudes areunity, so the transmit code is unit modulus. The corresponding arbitrarycomplex filter code of N_(r) samples is y(n), n=1, . . . , N_(r). Thesets of transmit and filter code samples are combined into vectors x andy, respectively. When calculating cross-correlations for N_(r)−N_(t)≥2,the transmit code is symmetrically zero-padded to be the same length asthe receive code such that

$\begin{matrix}{{\overset{\_}{x}(n)} = \left\{ {\begin{matrix}{0,{n = 1},\ldots\mspace{14mu},{\left( {N_{r} - N_{t}} \right)/2}} \\{{x\left( {n - {\left( {N_{r} - N_{t}} \right)/2}} \right)},{n = {{\left( {N_{r} - N_{t}} \right)/2} + 1}},\ldots\mspace{14mu},{\left( {N_{r} + N_{t}} \right)/2}} \\{0,{n = {{\left( {N_{r} + N_{t}} \right)/2} + 1}},\ldots\mspace{14mu},N_{r}}\end{matrix}.} \right.} & (2)\end{matrix}$Range response is the cross-correlation where only the N_(t)+N_(r)−1non-zero range samples are retained,

$\begin{matrix}\begin{matrix}{{r(k)} = \left\{ {\begin{matrix}{{\sum\limits_{n = {k + 1}}^{N_{r}}{{\overset{\_}{x}(n)}{y^{*}\left( {n - k} \right)}}},{k = 0},\ldots\mspace{14mu},{N - 1}} \\{{\sum\limits_{n = 1}^{N_{r} + k}{{\overset{\_}{x}(n)}{y^{*}\left( {n - k} \right)}}},{k = {{- N} + 1}},\ldots\mspace{14mu},{- 1}}\end{matrix},} \right.} \\{{= {\sum\limits_{n = {\max{({1,{k + 1}})}}}^{\min{({{N_{r} + k},N_{r}})}}{\overset{\_}{x}(n){y^{*}\left( {n - k} \right)}}}},{k = {{- N} + 1}},\ldots\mspace{14mu},{N - 1},}\end{matrix} & (3) \\{N = {\left( {N_{t} + N_{r}} \right)/2.}} & \;\end{matrix}$

The central mainlobe peak is at k=0.

The mainlobe equality constraint is a mainlobe response of unity, or,

$\begin{matrix}{{r(0)} = {{\sum\limits_{n = 1}^{N_{r}}{{\overset{\_}{x}(n)}{y^{*}(n)}}} = 1.}} & (4)\end{matrix}$

The nonlinear inequality constraint restricts SNR Loss to be less thanan acceptable number, β, usually around 3 dB.

$\begin{matrix}{{{SNR}\mspace{14mu}{Loss}} = {{\sum\limits_{n = 1}^{N_{t}}{{b(n)}^{2}{\sum\limits_{n = 1}^{N_{r}}{{y(n)}}^{2}}}} \leq {\beta.}}} & (5)\end{matrix}$

Initial values are random transmit waveform phases between −π and πradians and Gaussian complex receive waveform samples. The mainlobeconstraint of (4) can be satisfied for the initial iteration byrescaling the receive waveform by

$\begin{matrix}{\left. y_{0}\rightarrow\frac{y_{0}}{{r_{0}(0)}^{*}} \right.,} & (6)\end{matrix}$where x₀ and y₀ are the initial code vectors and r₀(0) is the mainlobevalue calculated with these codes in (4).

The sidelobe penalty function is a generalized Integrated Sidelobe Level(ISL),

$\begin{matrix}{J_{SL} = {\left\lbrack {\sum\limits_{k = {{- N} + 1}}^{- 1}{+ \sum\limits_{1}^{N - 1}}} \right\rbrack{w(k)}{{{r(k)}}^{2}.}}} & (7)\end{matrix}$

Flexibility in setting range weights allows, in example variations ofthe MOW process, suppressing certain high-clutter power rangesasymmetrically, ignoring ranges with no possible clutter, expanding themainlobe, and emphasizing sidelobes adjacent to the mainlobe for HRR.The standard ISL is for all w(k)=1, equally summing all sidelobe powerexcept for the mainlobe. For matched filters, the ISL function can bewritten in terms of Fourier transforms of the code, and iterationstypically update this transform. For mismatched codes, the transmitfunction is zero-padded before a Fourier transform, and (7) is typicallyevaluated in sample space to maintain the zero-padded properties of thetransmit function, but can optionally be evaluated in other domains.

In some implementations, the process for minimization of a nonlinearmulti-variable real scalar function with nonlinear constraints can usethe fmincon function available in MATLAB®. The penalty function andconstraints are invariant to a constant phase shift over all transmitand filter samples. Therefore the first transmit phase is assigned thevalue zero and is not part of the unknowns. The fmincon function usesonly real variables and function values, so a combined unknown vector oflength N_(t)+2N_(r)−1, is solved for,

$\begin{matrix}{{z = \begin{pmatrix}\overset{\sim}{\phi} \\{{Re}(y)} \\{{Im}(y)}\end{pmatrix}},{{\overset{\sim}{\phi}(n)} = {\phi\left( {n + 1} \right)}},{n = 1},\ldots\mspace{14mu},{N_{t} - 1.}} & (8)\end{matrix}$

The two non-linear equality constraints areRe[r(0)]=1,Im[r(0)]=0,  (9)and the single inequality constraint is (5).

For the matched case, an auto-correlation of the transmit code has thesymmetry r(k)=r*(−k), but this is not true in general forcross-correlations of the transmit and filter codes in the mismatchedcase. Even symmetric range weights produce asymmetric cross-correlationsdue to initial condition randomness. Symmetric range sidelobesr(k)=r(−k) result from symmetric codes. In this case, truncated firsthalf codes x_(s) and y_(s) are of lengths N_(tc)=(N_(t)+1)/2 andN_(rc)=(N_(r)+1)/2, respectively. Symmetrization is around the centralsample, or,

$\begin{matrix}{{\phi(n)} = \left\{ {\begin{matrix}{{\phi_{s}(n)},{n = 1},\ldots\mspace{14mu},N_{tc}} \\{{\phi_{s}\left( {N_{t} - n + 1} \right)},{n = {N_{tc} + 1}},\ldots\mspace{14mu},N_{t}}\end{matrix},{{y(n)} = \left\{ {\begin{matrix}{{y_{s}(n)},{n = 1},\ldots\mspace{14mu},N_{rc}} \\{{y_{s}\left( {N_{r} - n + 1} \right)},{n = {N_{rc} + 1}},\ldots\mspace{14mu},N_{r}}\end{matrix}.} \right.}} \right.} & (10)\end{matrix}$

Thus, the unknown vector of length N_(tc)+2N_(rc)−1 for symmetricwaveforms is

$\begin{matrix}{{z_{s} = \begin{pmatrix}{\overset{\sim}{\phi}}_{s} \\{{Re}\left( y_{s} \right)} \\{{Im}\left( y_{s} \right)}\end{pmatrix}},{{{\overset{\sim}{\phi}}_{s}(n)} = {\phi_{s}\left( {n + 1} \right)}},{n = 1},\ldots\mspace{14mu},{N_{tc} - 1.}} & (11)\end{matrix}$

Symmetric waveforms are used for the basic process as they have abouthalf the number of unknowns and therefore the search calculationconverges more rapidly. While symmetric codes are preferred, in somecases, asymmetric codes and range responses can be generated.

Results for an embodiment are shown in FIG. 3, 300, for a unit modulustransmit code with N_(t)=101 samples and varying number receive samplesN_(r) on the axis 301 between 101 and 501, incrementing by 10. All rangesamples except for the mainlobe are counted as sidelobes. The basicconstrained nonlinear minimization step 206 of flowchart 200 (shown inFIG. 2) minimizes the non-linear ISL penalty function (7), for theunknown vector in (11), with the non-linear equality constraint (9) andthe inequality constraint (5) to keep the SNR loss less than 3 dB. TheMATLAB® function fmincon is used with the interior point algorithm, atermination tolerance of 10⁻⁹ and a constraint tolerance of 10⁻¹⁰.

The CAN algorithm for minimizing ISL by a matched phase only code is runwith the Golomb waveform as an initial condition. The CAN Matlab sourcecodes were downloaded from http://www.sal.ufl.edu/book/. The convergencethreshold |x^((n))−x^((n-1))|, where the superscript is the iterationnumber, was reduced from 10⁻⁵ to 10⁻⁹.

In FIG. 3, the solid line with asterisk markers 302 is the CAN ISL usingfor the number of code samples the corresponding number on thehorizontal axis 301. The CAN ISL solid line with asterisks 302 decreasessmoothly but not monotonically from −16.4 dB at 101 samples to a minimumof −21.9 dB for 491 samples. Mean sidelobe improvement is somewhatcountered by an increase in the number of range samples summed. The MOWprocess improves performance. MOW ISL, represented with a solid line304, decreases from −21.9 dB at 101 receive samples to −68.3 dB at 491receive samples, in spite of having to sum over more range samples.

The CAN PSL, represented by a solid line with X markers 303, decreasesfrom −27.2 dB at 101 samples to a minimum of −42.7 dB at 481 samples.MOW PSL, represented by a dashed line 305, decreases from −35.5 dB at101 receive samples to −82.6 dB at 491 receive samples. PSL locationsoccur near the ends of the range extent.

Increasing the number of phase-only samples in the CAN algorithmimproves performance, but for a large number of samples, it is 46.4 dBworse for ISL and 39.9 dB worse for PSL when compared to the MOWprocess. The MOW process results shown in FIG. 3 always have 101transmit samples. As the transmit pulse width determines radar minimumrange, improving the CAN algorithm by increasing the number of samplesincreases radar minimum detection range, but for the MOW process, theincrease in receive waveform samples does not do this, as the transmitpulse width is constant.

Mean sidelobes of the MOW process are shown by a dash-dot line 306 andtracks ISL since the mean is calculated as ISL divided by one less thanthe number of range samples. It varies from 23 dB below ISL at 101receive samples to 27.8 dB below ISL at 501 receive samples. Mediansidelobes, shown by a dotted line 307 are lower still, reaching aminimum of −78.8 dB for 431 samples. Mean and median sidelobes aresometimes referred to as a sidelobe level for algorithms, such as NLFMchirps, that reduce sidelobes, but broaden the mainlobe.

Selected range responses are in shown FIG. 4. Range index k from (3) isdisplayed on the horizontal axis 401. The number of transmit samplesN_(t) is always 101. The mainlobe 402 at k=0 is constrained to 0 dB.Range responses are smoothed by a 5-point equal-weight mean forreadability. Responses within 3 samples of the mainlobe are notsmoothed. ISL and PSL are evaluated from unsmoothed data. The CANalgorithm, represented by a solid line with cross markers 403, has anISL of −16.4 dB and a PSL of −27.2 dB. Sidelobes away from the mainlobeare about −45 dB. The CAN range response has 201 samples. The number ofreceive samples for the MOW process starts at 101 and increases by 100to 501. For N_(r)=101, the same number as transmit samples, shown by thesolid line 404, the MOW process outperforms the CAN algorithm by 5.5 dBin ISL and by 8.3 dB in PSL. Range response for N_(r)=201 (line 405),301 (line 406), 401 (line 407), and 501 (line 408) show steady decreasesin ISL and PSL to spectacularly low values in the legend. Sidelobesnearest to the mainlobe are even lower. Higher clutter sidelobes awayfrom the mainlobe may have less of a deleterious effect on a radar asthey may be outside of the beamwidth, or have a very different radialvelocity and are separated by Doppler processing. A unit-modulusauto-correlation range response must have its power endpoints equal to1/N_(t) ² since for a single term, no cancellation is possible, whichfor 101 samples is −40.1 dB. This tracks with FIG. 4, line 403. MOWfilter codes taper in amplitude near the endpoints. As a result, all MOWsidelobe endpoints are much lower.

Multiple and continuously variable frequency chirps have loweredsidelobes but broadened mainlobes. Close in sidelobe level is extremelyimportant for minimum detectable velocity (MDV), separating closetargets, reducing clutter Doppler spread, and HRR radar. FIG. 5 showsthe first 10 unsmoothed sidelobes of the range response plot 400 of FIG.4. Range sidelobes are symmetric, so the legend covering the sidelobesleft of the mainlobes is not suppressing any information. Again, therange index k in 501 has the value 0 at the mainlobe. Mainlobe level in502 is still 0 dB. The CAN algorithm of 503 is the solid line with across marker 503 with a first sidelobe of −52.2 dB. MOW first sidelobevalues are in the legend, for receive samples numbers of 101 (solid line504), 201 (dashed line 505), 301 (dotted line 506), 401 (dash-dot line507), and 501 (solid line 508). The decrease is down to an amazing−141.0 dB for 501 filter samples.

Excluding additional range samples increases performance as shown inFIG. 6. Range response is plotted against the range index 601. Excludingp=0, 1, and 2 samples on both sides of the mainlobe for N_(t)=101,N_(r)=201, results in ISL of −37.9 (solid line 602), −42.4 (dashed line603), and −59.2 dB (dotted line 604), respectively. PSL is −52.7, −55.7,and −72.1 dB, respectively, where the region for the peak sidelobeexcludes the enlarged mainlobe. The insert 605 shows range samples nearthe mainlobe. As p increases, the mainlobe broadens, but the firstsidelobe level is lowest for the largest value of p. Although this isnot shown, increasing weight coefficients for a subset of ranges resultsin much lower sidelobes in those regions. In some implementations, theseweights can be asymmetric. Numerical experiments show that quantizationlosses at 12 bits are an acceptable few dB.

To maintain spectral containment to a target bandwidth, the MOW processcan, in some implementations, use oversampled transmit and receive codesthat are generated to minimize sidelobe power with constrained SNR lossand simultaneously minimize out of band spectral leakage. Usually, theinverse of the chip duration is equal to the frequency bandwidth. Herechips are oversampled by a factor L, implying τ_(s)=1/(LB), where B isthe bandwidth and τ_(s) is sample spacing. There are L samples per chipor inverse bandwidth. Therefore, the transmit code has N_(t) samples orN_(t)/L chips and similarly for the filter code. Since the mainlobe isoversampled, an additional L−1 samples on both sides of the mainlobe areexcluded from the minimization, as

$\begin{matrix}{J_{SLO} = {\left\lbrack {\sum\limits_{k = {{- N} + 1}}^{- L}{+ \sum\limits_{L}^{N - 1}}} \right\rbrack{w(k)}{{{r(k)}}^{2}.}}} & (12)\end{matrix}$

This oversampling implies that both codes should be bandlimited to[−0.5/L, 0.5/L] in units of cycles per sample, or equivalently, [−0.5,0.5] in units of cycles per chip. Spectral leakage to frequenciesoutside of this target bandwidth should be suppressed. In practice,there is a transition region between the target bandwidth and thesuppressed frequencies outside of the target bandwidth, and sidelobesare evaluated after the first notch from the mainlobe.

Band limiting is accomplished by adding weighted sums of penaltyfunctions for frequency bands. A set of N_(B) frequency bands in thetransmit code to avoid is identified, where the j^(th) band is betweenf_(j1) and f_(j2), with a corresponding weight w_(jT), and the penaltyfunction to minimize depends on the integrated spectral energy in theband of frequencies outside of the target bandwidth, as,

$\begin{matrix}{{{\sum\limits_{j = 1}^{N_{B}}{w_{jT}{\int_{f_{j\; 1}}^{f_{j\; 2}}{{df}{{\hat{X}(f)}}^{2}}}}} = {{x^{\dagger}R^{TR}x} = {\sum\limits_{m,{n = 1}}^{N_{t}}{{R^{TR}\left( {m,n} \right)}{b(m)}{b(n)}{\exp\left\lbrack {i\left( {{\phi(n)} - {\phi(m)}} \right)} \right\rbrack}}}}}\ {{R^{TR}\left( {m,n} \right)} = {\sum\limits_{j = 1}^{N_{B}}{w_{jT}\left\{ {\begin{matrix}{\frac{{\exp\left\lbrack {2\pi\;{i\left( {m - n} \right)}f_{j\; 2}\Delta\; t} \right\rbrack} - {\exp\left\lbrack {2\pi\;{i\left( {m - n} \right)}f_{j\; 1}\Delta\; t} \right\rbrack}}{2\pi\;{i\left( {m - n} \right)}\Delta\; t},{m \neq n}} \\{{f_{j\; 2} - f_{j\; 1}},\;{m = n}}\end{matrix},\mspace{79mu}{{\hat{X}(f)} = {\sum\limits_{n = 1}^{N_{t}}{{x(n)}{\exp\left\lbrack {{- 2}\pi\;{{if}\left( {n - 1} \right)}\Delta\; t} \right\rbrack}}}},} \right.}}}} & (13)\end{matrix}$

Frequency and chip spacing Δt have corresponding units.

The matrix R^(TR) is Hermitian Toeplitz, so the scalar product x^(†) R^(TR)x can be computed in O(N_(t)) operations. To better controlrelative penalty function magnitudes, a normalized form is used,

$\begin{matrix}{{J_{TR} = {x^{\dagger}{\overset{\_}{R}}^{TR}x}}{{{\overset{\_}{R}}^{TR}\left( {m,n} \right)} = {\frac{R^{TR}\left( {m,n} \right)}{R^{TR}\left( {1,1} \right)} + {\lambda^{TR}{{\delta\left( {m,n} \right)}.}}}}} & (14)\end{matrix}$

Diagonal loading by λ^(TR)=10⁻⁴ is used as the frequency penalty matrixis otherwise singular. Another method sums over discrete FFT frequencybins, but significant zero-padding for additional intermediate bins isrequired, or spectral power spikes up between the chosen bins.

An equivalent penalty function for the receive filter code is calculatedby

$\begin{matrix}{\mspace{79mu}{{{J_{RV} = {y^{\dagger}{\overset{\_}{R}}^{RV}y}},{{{\overset{\_}{R}}^{RV}\left( {m,n} \right)} = {\frac{R^{RV}\left( {m,n} \right)}{R^{RV}\left( {1,1} \right)} + {\lambda^{RV}{\delta\left( {m,n} \right)}}}}}\mspace{79mu}{{{\sum\limits_{j = 1}^{N_{B}}{w_{jR}{\int_{f_{j\; 1}}^{f_{j\; 2}}{{df}{{\hat{Y}(f)}}^{2}}}}} = {y^{\dagger}R^{RV}y}},\mspace{79mu}{{\hat{Y}(f)} = {\sum\limits_{n = 1}^{N_{r}}{{y(n)}{\exp\left\lbrack {{- 2}\pi\;{{if}\left( {n - 1} \right)}\Delta\; t} \right\rbrack}}}}}{{R^{RV}\left( {m,n} \right)} = {\sum\limits_{j = 1}^{N_{B}}{w_{jR}\left\{ {\begin{matrix}{\frac{{\exp\left\lbrack {2\pi\;{i\left( {m - n} \right)}f_{j\; 2}\Delta\; t} \right\rbrack} - {\exp\left\lbrack {2\pi\;{i\left( {m - n} \right)}f_{j\; 1}\Delta\; t} \right\rbrack}}{2\pi\;{i\left( {m - n} \right)}\Delta\; t},{m \neq n}} \\{{f_{j\; 2} - f_{j\; 1}},{m = n}}\end{matrix}.} \right.}}}}} & (15)\end{matrix}$

In a preferred implementation, and without loss of generality, frequencyband limits are the same as for the transmit code, as there is adifferent set of band weights. However, in some cases, the frequencyband limits may be different for the transmit code and the filter code.The R^(RV) matrix is also Hermitian complex.

The total objective function, sometimes referred to a penalty function,isJ=δ _(SL) J _(SLO)+δ_(TR) J _(TR)+δ_(RV) J _(RV).  (16)

Spectral leakage is defined as the power in the avoided frequenciesbands divided by power in the passband frequencies. Desired maximumleakage values are input (for example, as an input parameter 202 in theprocess represented by the flowchart 200). The minimization search of(16) is done in segments, with the output of one segment becoming theinitial value of the following segment. The frequency penalty functionweights are scaled at each segment by the ratio of the desired to actualleakage (for example, in step 209 of flowchart 200). Weighting forsidelobes is unchanged. Frequency penalty weight factors are notindependent, since if there is little transmit out-of-band spectralpower, there will be little from the filter code.

There is more flexibility with the asymmetric code. Additional frequencynotches may be inserted asymmetrically within the bandwidth. Forexample, if a particular nearby frequency channel needs strongersuppression, transmit code weights in that band can be increased.Conversely, if there is a strong emitter into a nearby band, filter codeband weights can be increased there.

Low-pass filtering necessary to contain transmit spectral power isincompatible with transmit waveforms ramping up at the first sample andramping down at the last pulse sample. Therefore, in someimplementations, the MOW process can ramp up and down the amplitude ofthe transmit code for a single chip duration using a Tukey taper, wherea trigonometric interpolation is used from the ends extending over Lsamples,

$\begin{matrix}{{{{x(n)} = {{b(n)}{\exp\left\lbrack {i\;{\phi(n)}} \right\rbrack}}},{n = 1},\ldots\mspace{14mu},N_{t}}{{b(n)} = \left\{ {\begin{matrix}{{\sin\left\lbrack \frac{\left( {n - 1} \right)\pi}{2\left( {L - 1} \right)} \right\rbrack},{n = 1},\ldots\mspace{14mu},L} \\{1,{n = {L + 1}},\ldots\mspace{14mu},{N_{t} - L}} \\{{\cos\left( \frac{\left( {n - N_{t} + L - 1} \right)\pi}{2\left( {L - 1} \right)} \right)},{n = {N_{t} - L + 1}},\ldots\mspace{14mu},N_{t}}\end{matrix}.} \right.}} & (17)\end{matrix}$

If available, the actual system ramping amplitude can be used instead.Except for this small fraction of amplitude ramping, the transmitwaveform is otherwise unit modulus.

Referring back to FIG. 1, once the generated transmit and filter codesare stored on the storage device 103 of the radar system 100,transmitter code output is then sent to the DAC 104. Referring to FIG.7A, spectra are shown for 5 additional samples between each sample tomodel DAC time dependence. Therefore, there are 40 total samples in eachchip, using an oversampling factor, L=8. Two DAC types are considered, azero order hold, with constant value until the next sample, and a firstorder hold with a linear phase transition. Other parameters areN_(t)=201, N_(r)=601, symmetric codes, and desired frequency suppressionof 20 dB. In the spectral plot FIG. 7A, frequency units are incycles/chip. Power spectral density (PSD) of the 8× oversampling ofchips 701 are shown, which extend between −4 and +4 cycles/chip. The DACspectra for zeroth-order hold 702 and for a first-order hold 703 arealso plotted. The vertical dashed-dot lines 704 are at −0.5 and 0.5cycles/chip and bracket the bandwidth. Discontinuities at chipboundaries (in value for zero hold DAC and in first derivative for firstorder hold DAC) cause spectral peaks at near 8 cycles/chip andharmonics. Low spectral leakage levels and the small DAC peaks show thata first order hold DAC with 8× oversampled samples produces a timewaveform without significant anomalies. Typically, CPM circuitry is notneeded due to the spectral containment effectiveness of the system;however, in some arrangements, CPM functionality may be employed.

Referring again to FIG. 1, the analog band-pass filter 105 is appliedbefore sending the signal to the amplifier 106 and the antenna 108.Since there is little out-of-band power, the bandpass filter transitionfrequencies can be placed well beyond the band limits 704, avoidingnonlinear phase response in this region. Frequency response of thefilter is generally better in the center of the band, where most of thepower is. Another advantage is that there is no straddling loss.

FIG. 7B shows the first order hold range response 705 of the transmitcode from FIG. 7A, and has low sidelobe levels. Zeroth order hold rangeresponse is very similar and is not shown. Dashed lines 706 demarcatethe expected mainlobe boundaries based on oversampling and filtering.

In some implementations, the MOW process can further account for Dopplertolerance. Chirp codes are fairly insensitive to Doppler shifts, butthis is not true for a polyphase code. Performance of the codescalculated here decreases for small Doppler frequencies. For matchedfilters and widely separated targets, a FFT across multiple pulsesyields a Doppler estimate, which is then used to correct received data.A bank of Doppler filters can be used and the one with lowest sidelobesis chosen. Generally, the goal here is to widen the Doppler frequencyrange where performance is good, defined as little loss in the mainlobewhile maintaining acceptably low sidelobes. The continuous timeambiguity function is defined as

$\begin{matrix}{{{\chi\left( {\tau,f} \right)} = {\int_{- \infty}^{\infty}\ {{{dtx}(t)}y*\left( {t - \tau} \right){\exp\left( {2\pi\;{ift}} \right)}}}},} & (18)\end{matrix}$

For discrete codes, (18) becomes

$\begin{matrix}{{\chi\left( {k,f} \right)} = \left\{ {\begin{matrix}{{\sum\limits_{n = {k + 1}}^{N_{r}}{{\overset{\_}{x}(n)}{\exp\left( {2\pi\;{ifn}} \right)}{y^{*}\left( {n - k} \right)}}},{k = 0},\ldots\mspace{14mu},{N - 1}} \\{{\sum\limits_{n = 1}^{N_{r} + k}{{\overset{\_}{x}(n)}{\exp\left( {2\pi\;{ifn}} \right)}{y^{*}\left( {n - k} \right)}}},{k = {{- N} + 1}},\ldots\mspace{14mu},{- 1}}\end{matrix},} \right.} & (19)\end{matrix}$where frequency f above has units of cycles/sample. The sidelobeminimization in (7) is extended to a set of Doppler frequencies,

$\begin{matrix}{{J_{SLD} = {\sum\limits_{m = 1}^{M}{{\overset{\sim}{w}\left( f_{m} \right)}{\sum\limits_{\underset{k \neq 0}{{k = {{- N} + 1}},}}^{N - 1}{{w(k)}{{\chi\left( {k,f_{m}} \right)}}^{2}}}}}},{{\chi\left( {0,0} \right)} = 1.}} & (20)\end{matrix}$

In some implementations, a single equality constraint leads to fastexecution times. In some implementations, the MOW process incorporatesthis Doppler tolerance into a joint optimization of both transmit andfilter codes, while simultaneously addressing frequency suppression, andSNR Loss control.

FIG. 8 is a flow chart of a computer device-implemented method 800 forstoring MOW transmit and filter codes in a storage device (e.g., storagedevice 103). The method 800 can comprise a step 801 for receiving one ormore input parameters that include a first parameter indicative of anumber of code samples of a first waveform, a second parameterindicative of a number of code samples of a second waveform, wherein thenumber of code samples of the second waveform is greater than the numberof code samples of the first waveform, and a third parameter indicativeof an oversampling ratio. These input parameters can correspond to inputparameters 202 and can further include parameters defining processvariations described in reference to FIG. 2.

The method 800 can comprise another step 802 generating, by one or moreprocessing devices executing a joint optimization process, (i) a firstsequence of values representing phase values for the first waveform, and(ii) a second sequence of values representing a set of complex amplitudevalues for the second waveform, the first and second sequences of valuesbeing generated such that an objective function of the first sequenceand the second sequence satisfies a set of one or more conditions,subject to one or more constraints, the objective function comprising aweighted sum of a Doppler tolerant metric indicative of a sidelobe levelof a simulated range response, a metric indicative of a frequencysuppression of the first waveform inside and/or outside of a targetbandwidth, and a metric indicative of a frequency suppression of thesecond waveform inside and/or outside of the target bandwidth. In somecases, the target bandwidth can be considered a target bandwidth. Insome cases, the joint optimization process of step 802 can beimplemented as a constrained nonlinear minimization 206 as described inreference to FIG. 2. The first sequence of values representing phasevalues for the first waveform can correspond to a series of phases for atransmit waveform such as transmit code 117. The second sequence ofvalues can correspond to a series of complex numbers for a filterwaveform such as filter code 118. In some cases, the objective functioncan be a weighted sum of multiple non-linear scalar functions such as(16), and the set of one or more conditions can correspond to a localminimization. In some cases, the metric indicative of a sidelobe levelof a simulated range response can correspond to J_(SLD) (see (20)). Insome cases, the metric indicative of a frequency suppression of thefirst waveform inside and/or outside of a target bandwidth cancorrespond to J_(TR) (see (14)). In some cases, the metric indicative ofa frequency suppression of the second waveform inside and/or outside ofthe target bandwidth can correspond to J_(RV) (see (15)).

Method 800 can further comprise a step 803 for storing the firstsequence of values and the second sequence of values in a storage deviceaccessible to a transmitter device and a receiver device such that ameasured range response to a transmit waveform transmitted by thetransmitter device based on the first sequence of values is computed bythe receiver device as a cross-correlation between (i) a sequence ofvalues representing a received signal, and (ii) the second sequence ofvalues. An example storage device may be storage device 103, accessibleby a transmitter device (e.g. transmitting system 115) and receiverdevice (e.g. receiving system 116). In some cases, a correlator (e.g.correlator 112) of the receiver device (e.g. receiving system 116) cancompute a measured range response as a cross-correlation between (i) asequence of values representing a received signal (e.g. returns receivedvia that antenna 108) and (ii) the second sequence of values (e.g.filter code 118).

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications can be made without departing fromthe spirit and scope of the processes and techniques described herein.In addition, the logic flows depicted in the figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In addition, other steps can be provided, or stepscan be eliminated, from the described flows, and other components can beadded to, or removed from, the described systems. Accordingly, otherembodiments are within the scope of the following claims.

What is claimed is:
 1. A computing device-implemented method comprising:receiving one or more input parameters that include a first parameterindicative of a number of code samples of a first waveform, a secondparameter indicative of a number of code samples of a second waveform,wherein the number of code samples of the second waveform is higher thanthe number of code samples of the first waveform, and a third parameterindicative of an oversampling factor; generating, by one or moreprocessing devices executing a joint optimization process, (i) a firstsequence of values representing phase values for the first waveform, and(ii) a second sequence of values representing a set of complex amplitudevalues for the second waveform, the first and second sequences of valuesbeing generated such that an objective function of the first sequenceand the second sequence satisfies a set of one or more conditions,subject to one or more constraints, the objective function comprising aweighted sum of a Doppler tolerant first metric indicative of a sidelobelevel of a simulated range response, a second metric indicative of afrequency suppression of the first waveform outside of a targetbandwidth, and a third metric indicative of a frequency suppression ofthe second waveform outside of the target bandwidth; and storing thefirst sequence of values and the second sequence of values in a storagedevice accessible to a transmitter device and a receiver device suchthat a measured range response to a transmit waveform transmitted by thetransmitter device based on the first sequence of values is computed bythe receiver device as a cross-correlation between (i) a sequence ofvalues representing a received signal, and (ii) the second sequence ofvalues.
 2. The method of claim 1, wherein the one or more inputparameters are received based on a set of target characteristicscomprising at least one of a target value for the first metricindicative of the sidelobe level of the simulated range response, atarget value for a fourth metric indicative of an overall frequencysuppression of the simulated range response, and a target value for afifth metric indicative of a Doppler tolerance.
 3. The method of claim1, wherein the metric indicative of the sidelobe level of the simulatedrange response comprises terms that represent at least one of anintegrated sidelobe level, a peak sidelobe level, and a median sidelobelevel with a range sample weighting.
 4. The method of claim 1, whereinthe one or more input parameters includes at least one of a sample ratesupported by the transmitter device and the receiver device, and atarget value for a metric indicative of an overall frequency suppressionof the simulated range response.
 5. The method of claim 1, wherein theone or more constraints comprises at least one of a target SNR lossthreshold; a constraint imposed on a mainlobe associated with thesimulated range response; and a first phase of the first sequence ofvalues constrained to
 0. 6. The method of claim 1, wherein weights ofthe weighted sum are adjusted periodically after an input number ofiterations until the objective function satisfies the set of one or moreconditions.
 7. The method of claim 1, wherein generating the firstsequence of values and the second sequence of values comprises:initializing a first set of decision variables as a series of phaseswithin a 2π range; initializing a second set of decision variables as aseries of complex values; and generating the first and second sequenceof values from the first and second set of decision variables,respectively, by executing the joint optimization process on the firstand second set of decision variables, the joint optimization processbeing constrained by the one or more constraints.
 8. The method of claim1, wherein at least one sidelobe of the simulated range response isexcluded from a computation of the objective function.
 9. The method ofclaim 1, wherein generating the first and second sequence of valuescomprises: generating multiple candidate sets of values, each candidateset comprising a candidate first sequence of values and a candidatesecond sequence of values; computing, for each candidate set, a value ofthe objective function; determining that a value of the objectivefunction for a particular candidate set is less than values ofcorresponding objective functions of other candidate sets; andresponsive to determining that the value of the objective functioncomputed for the particular candidate set is less than the values ofcorresponding objective functions of the other candidate sets, selectingthe first and second candidate sequence of values corresponding to theparticular candidate set as the first and second sequence of values,respectively.
 10. The method of claim 1, wherein the one or moreconditions comprises a condition that a value of the objective functionis locally minimized.
 11. The method of claim 1, wherein amplitudes ofthe transmit waveform are ramped up at a start of the transmit waveformand amplitudes of the transmit waveform are ramped down at an end of thetransmit waveform.
 12. The method of claim 1, wherein the Dopplertolerant metric indicative of a sidelobe level is a function of a timeambiguity function over a set of Doppler frequencies.
 13. A radar systemcomprising: a transmitter device configured to transmit a transmitsignal encoding a transmit waveform having a plurality of pulses, and areceiver device configured to receive a received signal, the transmitterdevice and the receiver device further configured to access a storagedevice configured to store a first sequence of values and a secondsequence of values such that a measured range response to the transmitsignal transmitted by the transmitter device based on the first sequenceof values is computed by the receiver device as a cross-correlationbetween (i) a sequence of values representing the received signal, and(ii) the second sequence of values, the first sequence of values and thesecond sequence of values generated by a computing device configured toexecute instructions to perform operations comprising: receiving one ormore input parameters that include a first parameter indicative of anumber of code samples of a first waveform, a second parameterindicative of a number of code samples of a second waveform, wherein thenumber of code samples of the second waveform is greater than the numberof code samples of the first waveform, and a third parameter indicativeof an oversampling factor; and generating, by one or more processingdevices executing a joint optimization process, (i) the first sequenceof values representing phase values for the first waveform, and (ii) thesecond sequence of values representing a set of complex amplitude valuesfor the second waveform, the first and second sequences of values beinggenerated such that an objective function of the first sequence and thesecond sequence satisfies a set of one or more conditions, subject toone or more constraints, the objective function comprising a weightedsum of  a Doppler tolerant first metric indicative of a sidelobe levelof a simulated range response,  a second metric indicative of afrequency suppression of the first waveform outside of a targetbandwidth, and  a third metric indicative of a frequency suppression ofthe second waveform outside of the target bandwidth.
 14. The radarsystem of claim 13, wherein the receiver device is further configured togenerate an output indicative of a location of an object as calculatedby multi-range processing around a peak range response.
 15. The radarsystem of claim 13, wherein the one or more input parameters arereceived based on a set of target characteristics, the set of targetcharacteristics comprising at least one of a target value for the metricindicative of the sidelobe level of the simulated range response, atarget value for a metric indicative of an overall frequency suppressionof the simulated range response, and a target value for a metricindicative of a Doppler tolerance.
 16. The radar system of claim 13,wherein the metric indicative of the sidelobe level of the simulatedrange response comprises terms that represent at least one of anintegrated sidelobe level, a peak sidelobe level, and a median sidelobelevel.
 17. The radar system of claim 13, wherein the one or more inputparameters includes at least one of a sample rate supported by thetransmitter device and the receiver device, and a target value for ametric indicative of an overall frequency suppression of the simulatedrange response.
 18. The radar system of claim 13, wherein the one ormore constraints comprises at least one of a target SNR loss threshold;a constraint imposed on a mainlobe associated with the simulated rangeresponse; and a first phase of the first sequence of values constrainedto
 0. 19. The radar system of claim 13, wherein weights of the weightedsum are adjusted periodically after an input number of iterations untilthe objective function satisfies the set of one or more conditions. 20.The radar system of claim 13, wherein generating the first sequence ofvalues and the second sequence of values comprises: initializing a firstset of decision variables as a series of phases within a 2π range;initializing a second set of decision variables as a series of complexvalues; and generating the first and second sequence of values from thefirst and second set of decision variables, respectively, by executingthe joint optimization process on the first and second set of decisionvariables, the joint optimization process being constrained by the oneor more constraints.
 21. The radar system of claim 13, wherein at leastone sidelobe of the simulated range response is excluded from acomputation of the objective function.
 22. The radar system of claim 13,wherein generating the first and second sequence of values comprises:generating multiple candidate sets of values, each candidate setcomprising a candidate first sequence of values and a candidate secondsequence of values; computing, for each candidate set, a value of theobjective function; determining that a value of the objective functionfor a particular candidate set is less than values of correspondingobjective functions of other candidate sets; and responsive todetermining that the value of the objective function computed for theparticular candidate set is less than the values of correspondingobjective functions of the other candidate sets, selecting the first andsecond candidate sequence of values corresponding to the particularcandidate set as the first and second sequence of values, respectively.23. The radar system of claim 13, wherein the one or more conditionscomprises a condition that a value of the objective function is locallyminimized.
 24. The radar system of claim 13, wherein amplitudes of thetransmit waveform are ramped up at a start of the transmit waveform andamplitudes of the transmit waveform are ramped down at an end of thetransmit waveform.
 25. The radar system of claim 13, wherein the Dopplertolerant metric indicative of a sidelobe level is a function of a timeambiguity function over a set of Doppler frequencies.
 26. One or morecomputer readable media storing instructions that are executable by aprocessing device, and upon such execution cause the processing deviceto perform operations comprising: receiving one or more input parametersthat include a first parameter indicative of a number of code samples ofa first waveform, a second parameter indicative of a number of codesamples of a second waveform, wherein the number of code samples of thesecond waveform is greater than the number of code samples of the firstwaveform, and a third parameter indicative of an oversampling factor;generating, by one or more processing devices executing a jointoptimization process, (i) a first sequence of values representing phasevalues for the first waveform, and (ii) a second sequence of valuesrepresenting a set of complex amplitude values for the second waveform,the first and second sequences of values being generated such that anobjective function of the first sequence and the second sequencesatisfies a set of one or more conditions, subject to one or moreconstraints, the objective function comprising a weighted sum of aDoppler tolerant first metric indicative of a sidelobe level of asimulated range response, a second metric indicative of a frequencysuppression of the first waveform outside of a target bandwidth, and athird metric indicative of a frequency suppression of the secondwaveform outside of the target bandwidth; and storing the first sequenceof values and the second sequence of values in a storage deviceaccessible to a transmitter device and a receiver device such that ameasured range response to a transmit waveform transmitted by thetransmitter device based on the first sequence of values is computed bythe receiver device as a cross-correlation between (i) a sequence ofvalues representing a received signal, and (ii) the second sequence ofvalues.